Computing power series in polynomial time

Computing Sums of Radicals in Polynomial Time

For sums of radicals P k i=1 i d i p i, where i; i are elements of some real algebraic number eld Q(); d i p i 2 R; we present a deterministic polynomial time algorithm to decide whether the sum is zero. The time is polynomial in the number of bits required to represent , the i's, i's and di's. The algorithm can be extended to sums of complex radicals over certain complex algebraic number elds.

Computing 2-Walks in Polynomial Time

A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3separators. This seminal result generalizes Tutte’s theorem that every 4-connected planar graph is Hamiltonian as ...

Diamond-alpha Polynomial Series on Time Scales

The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combinat...

Computing Leading Exponents of Noetherian Power Series

For a field k of characteristic zero, we study the field of Noetherian power series, k〈〈tQ〉〉, which consists of maps z : Q → k whose supports are Noetherian (i.e., reverse well-ordered) subsets of Q. There is a canonical valuation LE : k〈〈tQ〉〉 → Q ∪ {−∞} that sends a nonzero series to the maximum element of its support. Given a nonzero polynomial f(x, y) ∈ k[x, y] and a series z ∈ k〈〈tQ〉〉 that ...

Variations on computing reciprocals of power series